When President Barack Obama addresses the nation Monday evening, nine days after launching airstrikes against Libyan strongman Muammar Qadhafi, his challenge is to convince the public that, congressional nail-biting aside, he acted swiftly and decisively in the national interest.

He has his work cut out for him.

Congress has cried out in surround sound — with criticism coming from all corners — since U.S. missiles first struck Libyan targets. At best, the public is lukewarm about the mission: According to Gallup, 47 percent approve and 37 percent disapprove. And top Obama administration officials, some of whom have differed over the goals and motivation for the mission, will face tough questioning this week in open hearings and a private, classified briefing for members of Congress.

So when Obama takes the stage at the National Defense University at Fort McNair on Monday evening, there is a sense of urgency that he make an airtight case for his decision to intervene in Libya — and skeptics and supporters alike will also be looking for an endgame.

Here’s the narrative that Obama officials are selling in advance of the address: European and Arab-world allies begged America to get involved to prevent the slaughter of Libyan rebels; ignoring Qadhafi would embolden other despots to quash opponents by force; the United Nations and NATO are on board and the military and intelligence capabilities of the United States were crucial to setting up a “no-fly” zone. On top of that, even though Obama wants Qadhafi ousted, the American role will be wrapped up without an invasion, the officials say.

Or, as Rep. Jim Moran (D-Va.) put it to POLITICO on Sunday: “It was a bunch of bullies against a group of people who weren’t equipped to arm themselves, defend themselves. ... We stepped in and made it more of a fair fight. We could have just watched passively, but that is not who we are.”

House Speaker John Boehner’s camp said the public needs specifics, not platitudes.

“The American people need to hear the president outline a clear explanation of the scope and goal of our mission in Libya, the role America and our allies will play and what he regards as benchmarks for success,” Boehner spokesman Michael Steel said.

Obama likely gave a taste of what he’ll say Monday in his weekly address Saturday.

“We’re succeeding in our mission. ... Because we acted quickly, a humanitarian catastrophe has been avoided and the lives of countless civilians — innocent men, women and children — have been saved,” Obama said. “This military effort is part of our larger strategy to support the Libyan people and hold the Qadhafi regime accountable. Together with the international community, we’re delivering urgent humanitarian assistance. ... Muammar Qadhafi has lost the confidence of his people and the legitimacy to rule, and the aspirations of the Libyan people must be realized.”

But even Obama’s staunch allies are eager to hear more about the exit strategy for U.S. forces and Obama’s vision for who will run Libya in the near and long term.

“I just hope we don’t have a mission creep that moves from helping to avoid civilian casualties to engagement in a more significant military activity there. ... If we are again getting on a side in a civil war, that would be much more concerning to me,” Rep. Chaka Fattah (D-Pa.), who supports Obama’s action, told POLITICO on Sunday. “I’d like to have more definition on that and where we are post our involvement in Libya. ... What is it that we would like to see?”

The U.S. intervention in Libya is the biggest test so far of Obama’s ability to present himself as a strong and shrewd commander in chief of the American military. This is the first major operation begun during his presidency, and his explanation of it will provide a glimpse of his vision for how the president should plan, execute and conclude a military campaign. And if it works, it’s a political masterstroke that could set him apart from President George W. Bush, whose long, confident strides into war became longer slogs into nation-building. It could also provide contrast with his Democratic predecessor, Bill Clinton, who was criticized for waiting too long before using American might, particularly in the Balkans.

But there’s risk, too. The hostilities in Libya haven’t abated. A campaign the White House promised would be measured in “days, not weeks” is now into Week Two. The fundamental mission of the U.S. and its allies appears to have moved past establishing a “no fly” zone to prevent Qadhafi from advancing and morphed into a second-phase effort to assist the rebels in pushing westward.

No one has clearly articulated a vision for what Libya would look like at the end of the U.S. engagement there.

“If it ends and Qadhafi is still sitting there, being able to say with some credibility that he has just fought the mother of all battles, ... you bet it will be damaging to our country,” Donald Rumsfeld, who served as defense secretary for Presidents Gerald Ford and George W. Bush, said on ABC’s “This Week.”

While Obama has said that the scope and duration of the engagement will be limited, even his defense secretary, Robert Gates, can’t specify what that means.

“I don’t think anybody knows the answer,” Gates said on “This Week,” when anchor Jake Tapper asked whether U.S. forces would still be engaged in Libya at the end of this year. He also told NBC’s David Gregory on “Meet the Press” that “I don’t think [Libya’s] a vital interest for the United States,” though he explained there are greater regional interests at play for America.

Gates’s answers speak to Obama’s difficulties on Capitol Hill, where many lawmakers are unsure of the aims, costs, duration and wisdom of the mission. The military action comes as Congress is skidding toward yet another budget showdown over how deeply to cut spending, further complicating the White House relationship with Capitol Hill.

“This is going to get so ugly on the Hill. They have no idea how this is going to end,” a senior Senate Democratic aide told POLITICO after Gates’s comments. “If the administration can’t define the national interest, how can senators define it to their constituents?”

But Obama is winning robust support from a bipartisan set of Senate hawks who were calling for U.S. intervention before it happened, as well as from Democratic leaders in the Senate and some rank-and-file Democrats in both chambers.

Democratic doves have been chagrined at the adventurism of their president, who has added a third Muslim-world front to the Iraq and Afghanistan wars they abhor. The Republican speaker and colleagues on both sides of the aisle have kept up a steady drumbeat of questions about the cost, duration, scope and goals of the mission. So have some centrist senators, including Democrat Jim Webb of Virginia and Indiana’s Dick Lugar, who is the top Republican on the Senate Foreign Relations Committee.

A handful of liberal and conservative lawmakers have questioned whether Obama overstepped his authority under the War Powers Act by using military force without a clear, imminent threat to the United States.

Fattah called that argument “nonsense,” noting that Congress could have ended its own recess to act on the Libya mission had Republican and Democratic leaders chosen to do so.

And Moran, a member of the House Defense Appropriations Subcommittee, said congressional hand-wringing underscores the need for the president to have the authority to deploy forces quickly.

“That’s exactly why the executive branch needs to simply act to do the right thing,” Moran said. “Too many members of the Congress are too easily influenced by public opinion. I think the congressional response has been disappointing.”